Motion in a plane (quick revision formula)

Motion in a plane refers to the movement of an object in two dimensions, typically represented by the x and y axes. It involves the analysis of displacement, velocity, and acceleration in both the horizontal and vertical directions. Understanding the basic formulas and concepts related to motion in a plane is crucial for solving problems in physics. Here, we will discuss some important formulas and concepts to revise this topic quickly.

Displacement:
- Displacement is a vector quantity that represents the change in position of an object.
- It is given by the formula:
Δr = r₂ - r₁
where Δr is the displacement, r₂ is the final position vector, and r₁ is the initial position vector.

Velocity:
- Velocity is a vector quantity that represents the rate of change of displacement.
- Average velocity is given by the formula:
v = Δr / Δt
where v is the average velocity, Δr is the displacement, and Δt is the change in time.
- Instantaneous velocity is given by the formula:
v = dr / dt
where v is the instantaneous velocity, dr is an infinitesimally small displacement, and dt is an infinitesimally small change in time.

Acceleration:
- Acceleration is a vector quantity that represents the rate of change of velocity.
- Average acceleration is given by the formula:
a = Δv / Δt
where a is the average acceleration, Δv is the change in velocity, and Δt is the change in time.
- Instantaneous acceleration is given by the formula:
a = dv / dt
where a is the instantaneous acceleration, dv is an infinitesimally small change in velocity, and dt is an infinitesimally small change in time.

Projectile Motion:
- Projectile motion refers to the motion of an object projected into the air, under the influence of gravity.
- In the absence of air resistance, the horizontal and vertical motions are independent of each other.
- The horizontal velocity remains constant throughout the motion, while the vertical velocity changes due to the acceleration due to gravity.
- The range of projectile motion is given by the formula:
R = (v₀² * sin(2θ)) / g
where R is the range, v₀ is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

Circular Motion:
- Circular motion refers to the motion of an object along a circular path.
- The centripetal acceleration of an object in circular motion is given by the formula:
aₙ = v² / r
where aₙ is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circular path.
- The centripetal force required to keep an object in circular motion is given by the formula:
Fₙ = m * aₙ
where Fₙ is the centripetal force, m is the mass of the object, and aₙ is the centripetal acceleration.

These are some of the important formulas and concepts related to motion in a plane. Understanding and applying these formulas can help in solving problems and analyzing the motion of objects in a two-dimensional space.